Sightseeing:

What is the difference between temples and shrines?
Temples are like "churches" for buddhism and shrines for Shinto.

To many of us Japanese people(at least to my family and to all my friends), Shinto is not a religion.
To me, the most important teaching in Shinto is "thanking everything."
This is not a religion. It is a teaching, it is common sense.

There is a spirit in each and all existing things, in flowers, in animals, in insects, in rocks, in houses, in humans, etc. So thank them. Thenk vegetables and animals when they are served on a plate. Thank your house when it rains. Thank rain for giving us water. Etc. Easy and happy teaching.

Shinto is technically, a mythology that explains the origin of the Imperial family. However, to be honest, I do not think of the Imperial family when I visit shrines. This is a very delicate topic. I will choose my words carefully. I am a proud Japanese. I am grateful that the Emperor is the symbol(it is stated in the Constitution of Japan that the Emperor is the "symbol" of Japan) of Japan. More than a symbol. However, I think of shrines and the Imperial family almost separately.
*By the way, the Constitusion of Japan is almost solely created by GHQ.

To many of us Japanese people(again, for me and people around me), Buddhism is more like religion.
However, Japanese buddhism is very relaxed. Not strict at all(for example, well over 90% of the monks eat fish/meat). They cannot be strict so they can co-exist with another "religion."
About 1,400 years agao from now, Prince Shoutoku, one of the smartest guys every in the history of Japan, divided the power of Buddhism and Shinto equally so they can co-exist without having hatred.

Even now, Shinto takes care of birth ceremonies and Buddhism takes care of death ceremonies(funerals). Money is divied, power is divided, therefor, no disputes.
*However, there are disputes between different branches of Buddhism. Just like all the other religions.

Now, let's take a look at temples and shrines that I think is most worth visiting!
*My aim here is NOT to make the LONGEST list. I made a list of the places that I think is truly worth visiting.

Temples:

Kiyomizu temple- This continues to be my number 1 temple. The big balcony(Kiyomizu no butai) is beautiful to look at and you can also see the most beautiful view of the city from there. One of the very rare temples to open early, so it is a great place to start your sightseeing day without wasting morning time.

Golden Pavilion- Kinkakjuji is the Japanese name of the temple. This "symbol of wealth" temple has been around for about 600 years. However, Kinkakuji that you see now was rebuilt in 1950's after it was burned down by an apprentice monk. The monk commited suicide after burning the temple. His mother came to Kyoto(police asked her to come) and on her way back home, she also commited suicide.
The temple is definitely worth a visit. I used to not like it when I was in my early teens, as I could go neither close to the temple nor go inside. Now I appreciate it's beautiful and calm posture.

Nishi Honguwanji-Headquarters of one of the larger branches of Japanese buddhism, Joudo-shinshu. When Oda Nobunaga tried to rule Japan, he saw Honguwanji as one of most powerful influences over the people. In 2011 and 2012, they were celebrating the 750th anniversary of the founder's passing.
This is one of my highest recommendations for temples. The big tatami room with hundreds of tatami mats is relaxing and gorgeous. Small children will enjoy walking and running around in this temple. My daughter did, when she was 1 and a half. Also, there is not entrance fee.

Daitokuji- One of the head zen temples. This is the place to go if you like gardens. Of course, you want to check their zen gardens.. However, each temple(there are many temples within Daitokuji) you need to pay the entrance fee.

Tofukuji- Also one of the head zen tepmles. This place also with a beautiful garden. If you are in Kyoto in November, you must go see this temple. The color of the leaves here is amazing.

Chionin- Home of Honen, the founder of Jodo branch of buddhism. If you visit Chionin, try to find the seven wonders of Chionin.


Shrines:

Fushimi Inari- This is a famous one, but still worth a visit. The shrine is most famous for better success in business. The red torii-gates are all donations from business owners. The head shrine of all the 40,000 Inari shrines all across the nation. No wonder there are many donation gates.

Shimogamo shrine- The guardian shrine of Kyoto. One of the oldest shrines/temples in Japan. When it was built is uncertain. There is a writing that states "repairing the gate" in 90 BC. Many thinks this is probably around the time they built it. So about 2,100years ago. Tadasuno mori forest is just as famous as the shrine(the forest is in the shrine's property). The forest is natural, but taken care of very well. There are a ccouple of hundreds of trees that are between 200-600years old.

Yasaka shrine- Shrine for the common people of Kyoto. Gion festival is centered around this shrine.


Others:

Philosopher's path- A beautiful path along the canal from Ginkakuji down to Nanzenji in the north east Kyoto. The famous Japanese philosopher, Nishida Kitaro used to walk here, therefor it is called the philosopher's walk(or path). The walk is very beautiful in the fall and spring with autumn leaves and cherry blossoms, respectively.

Mount Hiei(Enryakuji temple)- Saicho, one of the two greatest Japanese monks(along with Kukai) of all time is the founder of the temple. Kukai's Mount Koya is now becoming a big tourist spot for international tourists, but for many Japanese people, Saicho's Mount Hiei(Enfyakuji temple) is on the same level, possibly more famous(this is debatable). Maybe I say this because, my wife is from Enryakuji-gakuei Mt. Hiei High school. It is a beautiful view of up on the mountain. Lake Biwa(the largest lake in Japan) on the west, Kyoto(the old national capital) on the east.

Aoibashi bridge- View from this bridge is beautiful in the cherry blossom season. Sakura(cherry trees) is lined up along the Kamo river and no tall buildings around. You can see the mountain range in the back ground and to your right, you can see the Daimonji mountain, which is lit up on fire for Obon festival.
This is my favorite place to view sakura, out of all the other places in Kyoto. My best recommendation for the "best place to see sakura in Kyoto"